University of Leicester Doctoral Thesis Betting Markets: Defining odds restrictions, exploring market inefficiencies and measuring bookmaker solvency Author: Supervisor: Dominic Cortis Professor Jeremy Levesley A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy in the Actuarial Science Research Group Department of Mathematics May 2016 Declaration of Authorship I, Dominic Cortis, declare that this thesis titled, ‘Betting Markets: Defining odds restrictions, exploring market inefficiencies and measuring bookmaker sol- vency’ and the work presented in it are my own. I confirm that: This work was done wholly or mainly while in candidature for a research degree at this University. Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated. Where I have consulted the published work of others, this is always clearly at- tributed. Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. I have acknowledged all main sources of help. Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself. Signed: Date: i No time will be better Family, exams, work, friendships, romance Fiver on league win for Leicester city Bond, Share, Derivative, Swap, annuity for term or life, in arrears or advance Poker, life, lottery, betting, finance Certain, Likely, impossible, unlikely Outcome and result unknown precisely We all get a turn in the games of chance Decisions to take: fast? for a lifetime? Action forever we cannot adjourn But most surely it cannot be a crime To stop, ask, calculate, think, trust, test, doubt Therefore we bite the time to take our turn Yet seldom we win a chance to turn time. UNIVERSITY OF LEICESTER Abstract College of Science and Engineering Department of Mathematics Doctor of Philosophy Betting Markets: Defining odds restrictions, exploring market inefficiencies and measuring bookmaker solvency by Dominic Cortis Betting markets have been of great interest to researchers as they represent a simpler set-up of financial markets. With an estimated Gross Gambling Revenue of 45bn yearly on betting on outcomes alone (excluding other gambling markets such as Casino, Poker and Lottery), these markets deserve attention on their own merit. This thesis provides simple mathematical derivation of a number of key statements in set- ting odds. It estimates the expected bookmaker profit as a function of wagers placed and bookmaker margin. Moreover it shows that odds set by bookmakers should have implied probabilities that add up to at least one. Bookmakers do not require the exact probabil- ity of an outcome to have positive expected profits. They can increase profitability by having more accurate odds and offering more multiples/accumulators. Bookmakers can lower variation in payouts by maintaining the ratio of wagers and implied probability per outcome. While bookmakers face significant regulatory pressures as well as increased taxes and levies, there is no standard industry model that can be applied to evaluate the mini- mum reserves for a bookmaker. Hence a bookmaker may be under/over-reserving funds required to conduct business. A solvency regime for bookmakers is presented in this work. Furthermore a model is proposed to forecast soccer results – this focuses on goal differ- ences in contrast to traditional models that predict outcome or goals scored per team. Significant investigations are made on the inefficiencies of betting markets. The likelihood of Brazil reaching different stages of the 2014 World Cup, as perceived by odds, is compared to events on and outside the pitch to imply bias. An analysis of over 136,000 odds for European soccer matches shows evidence of the longshot bias. Finally this work investigates how it is possible to profit from market inefficiencies on betting exchanges during short tournaments by using a Monte Carlo simulation method as a quasi-arbitrage model. Acknowledgements I thank all those who helped me at the start, during or at the end of this journey by being behind me to push me, in front of me to show me the way, below me to hold me when I fell, above me to keep an eye out or beside me to encourage me. v Contents Declaration of Authorshipi Abstract iii Acknowledgementsv Contents vi List of Figures ix List of Tablesx 1 Introduction1 2 Expected values and variances in bookmaker payouts: A Theoretical approach towards setting limits on odds5 2.1 Evaluating Odds................................5 2.2 Bookmaker Margin...............................7 2.3 Outcomes on a Single Event..........................8 2.3.1 Restrictions on Setting Odds.....................8 2.3.2 Bookmaker Profit and Deviation...................9 2.4 Multiples.................................... 12 2.5 Conclusion.................................... 15 3 Developing a solvency framework for bookmakers 16 3.1 Bookmaker’s Portfolio............................. 17 3.2 Bundling Technique.............................. 18 3.2.1 Variance per Bundle.......................... 18 3.2.2 Portfolio Variance........................... 19 3.3 Determining Solvency............................. 21 3.4 Challenges and Extensions........................... 22 3.5 Conclusion.................................... 23 4 A public (mis)interpretation of Brazil’s world cup performance 24 4.1 Introduction................................... 24 vi Contents vii 4.2 Context: The World Cup Finals and Betting Markets............ 25 4.2.1 World Cup Finals............................ 25 4.2.2 World Cup Betting Markets...................... 26 4.3 Data and Trends................................ 27 4.3.1 Data................................... 27 4.3.2 Perception versus Performance.................... 27 4.3.2.1 Group Stage – Croatia [3-1], Mexico [0-0], Cameroon [4-1] 27 4.3.2.2 Knock-out Stage – Chile [1-1], Colombia [2-1], Germany [1-7].............................. 30 4.4 Discussion and Conclusion........................... 30 5 Predicting goal difference 33 5.1 Introduction................................... 33 5.2 Literature Review................................ 34 5.3 Data and methodology............................. 36 5.4 A four-factor linear model........................... 38 5.5 Draws estimated using m=0.25 ........................ 40 5.6 Conclusion and Discussion........................... 42 6 Betting market efficiency in European soccer 44 6.1 Introduction................................... 44 6.2 Literature Review................................ 45 6.2.1 The Favourite-Longshot Bias..................... 45 6.2.2 Exploring Other Biases........................ 48 6.2.3 Testing for Insider Knowledge..................... 49 6.2.4 Arbitrage Opportunities........................ 50 6.3 Methodology.................................. 51 6.3.1 Data................................... 51 6.3.2 Determining Probabilities and Other Metrics............ 52 6.3.3 Subdividing the ranges......................... 52 6.3.4 Correlations............................... 53 6.4 Analysis and Results.............................. 54 6.4.1 General................................. 54 6.4.2 Conflicting results in Overs Market.................. 55 6.5 Conclusion.................................... 57 7 Profiting on inefficiencies in betting derivative markets: The case of UEFA EURO 2012 61 7.1 Introduction................................... 61 7.2 Betting Markets................................. 63 7.3 Method..................................... 64 7.3.1 The Algorithm............................. 64 7.3.2 Comparing Output with Market................... 66 7.4 Results...................................... 68 7.5 Conclusion.................................... 68 8 Conclusion 70 8.1 The Business of Bookmaking......................... 70 Contents viii 8.2 Predicting Soccer outcomes.......................... 72 8.3 (In)efficient markets.............................. 72 8.4 Other Work................................... 73 8.5 Implications on the Financial Industry.................... 73 A Payout Distribution for Constant Bookmaker Margin 75 B Pricing risk through simulation: Revisiting Tilley bundling and least square Monte Carlo methods 76 B.1 Motivation.................................... 76 B.2 Introduction................................... 76 B.2.1 Use of Monte Carlo methods in pricing derivatives......... 77 B.2.2 Risk profiles............................... 78 B.2.3 European-style Option in the GBM one-factor model........ 78 B.3 The Two Monte Carlo methods........................ 80 B.3.1 Steps Involved............................. 80 B.3.2 A Numerical Example of Pricing: An American Call Option.... 81 B.3.2.1 LSM.............................. 82 B.3.2.2 Tilley Bundles........................ 84 B.3.2.3 Comparison of the two systems............... 85 B.3.3 A Numerical Example of Risk Pricing: An American Put Option. 87 B.3.4 Comparing Results........................... 89 B.4 Future Work.................................. 89 C Data Adjustments 90 D Comparison of Weibull Parameters with past Euro and African Cup of Nations Games 92 D.1 Introduction................................... 92 D.2 Data....................................... 92 D.3 Data Treatment................................. 93 D.4 Results...................................... 94 E Sample Output for Group D 95 References